From Eq. (REF ), we can identify several branching points associated with how different fields analyze quasicrystals. For example, we can either (i) study directly the ab-initio atomic arrangement and hybridization as discussed above, (ii) analyze the most general spectral properties that can be obtained from a specific quasiperiodic atomic arrangement, i.e., without turning to ab-initio extraction of parameters; this is similar to symmetry group analysis of spectral features of periodic crystals, and (iii) analyze specific sub-cases of the general case (REF ), e.g., by considering solely short-ranged potentials. In the spirit of (ii), much research is pursued within the mathematical physics community, aiming to answer (a) whether the spectrum has a well-defined structure with a countable number of bands and gaps [see, e.g., Ref. [1]}], (b) whether the eigenstates of the quasicrystals are spatially-extended or localized [see, e.g., Refs. [2]}, [3]}, [4]}], and (c) whether a topological classification of spectral bands of the quasicrystals is possible by harnessing K-theoretical tools [see e.g., Refs. [1]}, [6]} and references therein]. Note that (a) and (c) are often combined, as specific quasiperiodic sequences fix the integrated density of states as a topological index that allows for labeling of spectral gaps of the material, and thus their enumeration [1]}. In the context of (iii), different approximations are taken, such as assuming only short-ranged nearest-neighbor hopping (\(t^{\phantom{\dag }}_{m\ne 1}=\lambda ^{\text{od}}_{m\ne 1}=0\) ), on top of which only diagonal (\(t^{\phantom{\dag }}_{1}=\lambda ^{\text{od}}_{1}=0\) ) or off-diagonal (\(\lambda ^{\text{d}}=0\) ) models are explored. Note that the large variety of such approximations in dimensions larger than 1 leads to different models exhibiting different effects, while describing the same long-ranged quasiperiodic spatial modulation.
